# Linear algebra efficient way for inconsistent systems

I am quite new in linear algebra and learned that a system is inconsistent when two planes or lines are parallel to each other. I wondered how to find inconsistent systems very fast using this idea, and taught if I could find at least two planes or lines parallel to each other, then the system must be inconsistent, therefore, no solutions. However, I found out from my recent question practices that sometimes even though two planes are not parallel to each other, I can still get an inconsistent system. I believe the cause of this is that there are also inconsistent systems when there is a triangular tube made by the planes. However now my question is, how can I determine that a set of plane equations indicate a triangular tube ? (I know that an inconsistent system can be found using row-reduction, but I want to find faster ways of doing so)

Most of my ideas are coming from this video (you will also see the info about the triangular tube there): https://www.youtube.com/watch?v=EzHQCq-jIl0

I would appreciate any help, thank you.

• Row reduction is about as fast as you are going to get for a general set of linear equations.
– Paul
May 4, 2021 at 15:09